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BEGIN:VEVENT
SUMMARY:Dang-Khoa Nguyen (University of Calgary)
DTSTART:20220926T180000Z
DTEND:20220926T190000Z
DTSTAMP:20260404T095423Z
UID:NTC/1
DESCRIPTION:Title: Height gaps for coefficients of D-finite power series\nby Dang-Khoa
Nguyen (University of Calgary) as part of Lethbridge number theory and co
mbinatorics seminar\n\nLecture held in University of Lethbridge\, room M10
40 (Markin Hall).\n\nAbstract\nA power series $f(x_1\,\\ldots\,x_m)\\in \\
mathbb{C}[[x_1\,\\ldots\,x_m]]$ is said to be D-finite if all the partial
derivatives of $f$\n span a finite dimensional vector space over\n the fie
ld $\\mathbb{C}(x_1\,\\ldots\,x_m)$. For the univariate series $f(x)=\\sum
a_nx^n$\, this is equivalent to the condition that the sequence $(a_n)$ i
s P-recursive meaning a non-trivial linear recurrence relation of the form
:\n $$P_d(n)a_{n+d}+\\cdots+P_0(n)a_n=0$$\n where the $P_i$'s are polynomi
als. In this talk\, we consider D-finite power series with algebraic coeff
icients and discuss the growth of the Weil height of these coefficients.\n
\n \n This is from a joint work with Jason Bell and Umberto Zannier in 2
019 and a more recent work in June 2022.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hugo Chapdelaine (Université Laval)
DTSTART:20221031T180000Z
DTEND:20221031T190000Z
DTSTAMP:20260404T095423Z
UID:NTC/2
DESCRIPTION:Title: Computation of Galois groups via permutation group theory\nby Hugo
Chapdelaine (Université Laval) as part of Lethbridge number theory and co
mbinatorics seminar\n\n\nAbstract\nIn this talk we will present a method t
o study the Galois group of certain polynomials defined over $\\Q$.\nOur a
pproach is similar in spirit to some previous work of F. Hajir\, who studi
ed\, more than a decade ago\, the generalized Laguerre polynomials using a
similar approach.\nFor example this method seems to be well suited to stu
dy the Galois groups of Jacobi polynomials (a classical family of orthogon
al polynomials with two parameters --- three if we include the degree). Gi
ven a polynomial $f(x)$ with rational coefficients of degree $N$ over $\\Q
$\, the idea consists in finding a good prime $p$ and look at the Newton p
olygon of $f$ at $p$. Then combining the Galois theory of local field over
$\\Q_p$ and some classical results of the theory of permutation of groups
we sometimes succeed in showing that the Galois group of $f$ is not solva
ble or even isomorphic to $A_N$ or $S_N$ ($N\\geq 5$).\n\nThe existence of
a good prime $p$ is subtle. In order to get useful results one would need
to have some "effective prime existence results". As an illustration\, we
would like to have an explicit constant $C$ (not too big) such that for a
ny $N>C$\, there exists a prime $p$ in the range $N < p < \\frac{3N}{2}$ s
uch that\ngcd$(p-1\,N)= 1 \\text{ or } 2$ (depending on the parity of $N$)
. Such a result is not so easy to get when $N$ is divisible by many distin
ct and small primes. We hope that such effective prime existence results a
re within the reach of the current techniques used in analytic number theo
ry.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julie Desjardins (University of Toronto)
DTSTART:20221117T210000Z
DTEND:20221117T220000Z
DTSTAMP:20260404T095423Z
UID:NTC/3
DESCRIPTION:Title: Torsion points and concurrent lines on Del Pezzo surfaces of degree one
\nby Julie Desjardins (University of Toronto) as part of Lethbridge nu
mber theory and combinatorics seminar\n\n\nAbstract\nThe blow up of the an
ticanonical base point on X\, a del Pezzo surface of degree 1\, gives rise
to a rational elliptic surface E with only irreducible fibers. The sectio
ns of minimal height of E are in correspondence with the 240 exceptional c
urves on X. A natural question arises when studying the configuration of t
hose curves : \n\nIf a point of X is contained in "many" exceptional curve
s\, is it torsion on its fiber on E?\n\nIn 2005\, Kuwata proved for del Pe
zzo surfaces of degree 2 (where there is 56 exceptional curves) that if "m
any" equals 4 or more\, then yes. In a joint paper with Rosa Winter\, we p
rove that for del Pezzo surfaces of degree 1\, if "many" equals 9 or more\
, then yes. Moreover\, we find counterexamples where a torsion point lies
at the intersection of 7 exceptional curves.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathieu Dutour (University of Alberta)
DTSTART:20221128T190000Z
DTEND:20221128T200000Z
DTSTAMP:20260404T095423Z
UID:NTC/4
DESCRIPTION:Title: Theta-finite pro-Hermitian vector bundles from loop groups elements
\nby Mathieu Dutour (University of Alberta) as part of Lethbridge number t
heory and combinatorics seminar\n\nLecture held in University of Lethbridg
e\, room M1040 (Markin Hall).\n\nAbstract\nIn the finite-dimensional situa
tion\, Lie's third theorem provides a correspondence between Lie groups an
d Lie algebras. Going from the latter to the former is the more complicate
d construction\, requiring a suitable representation\, and taking exponent
ials of the endomorphisms induced by elements of the group.\n\nAs shown by
Garland\, this construction can be adapted for some Kac-Moody algebras\,
obtained as (central extensions of) loop algebras. The resulting group is
called a loop group. One also obtains a relevant infinite-rank Chevalley l
attice\, endowed with a metric. Recent work by Bost and Charles provide a
natural setting\, that of pro-Hermitian vector bundles and theta invariant
s\, in which to study these objects related to loop groups. More precisely
\, we will see in this talk how to define theta-finite pro-Hermitian vecto
r bundles from elements in a loop group. Similar constructions are expecte
d\, in the future\, to be useful to study loop Eisenstein series for numbe
r fields.\n\nThis is joint work with Manish M. Patnaik.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Florea (University of California - Irvine)
DTSTART:20221205T190000Z
DTEND:20221205T200000Z
DTSTAMP:20260404T095423Z
UID:NTC/5
DESCRIPTION:Title: Negative moments of the Riemann zeta-function\nby Alexandra Florea
(University of California - Irvine) as part of Lethbridge number theory an
d combinatorics seminar\n\n\nAbstract\nI will talk about recent work towar
ds a conjecture of Gonek regarding negative shifted moments of the Riemann
zeta-function. I will explain how to obtain asymptotic formulas when the
shift in the Riemann zeta function is big enough\, and how we can obtain n
on-trivial upper bounds for smaller shifts. I will also discuss some appli
cations to the question of obtaining cancellation of averages of the Mobiu
s function. Joint work with H. Bui.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Debanjana Kundu (University of British Columbia)
DTSTART:20221003T180000Z
DTEND:20221003T190000Z
DTSTAMP:20260404T095423Z
UID:NTC/6
DESCRIPTION:Title: Studying Hilbert's 10th problem via explicit elliptic curves\nby De
banjana Kundu (University of British Columbia) as part of Lethbridge numbe
r theory and combinatorics seminar\n\nLecture held in University of Lethbr
idge: M1040 (Markin Hall).\n\nAbstract\nIn 1900\, Hilbert posed the follow
ing problem: "Given a Diophantine equation with integer coefficients: to d
evise a process according to which it can be determined in a finite number
of operations whether the equation is solvable in (rational) integers."\n
\nBuilding on the work of several mathematicians\, in 1970\, Matiyasevich
proved that this problem has a negative answer\, i.e.\, such a general `pr
ocess' (algorithm) does not exist.\n\nIn the late 1970's\, Denef--Lipshitz
formulated an analogue of Hilbert's 10th problem for rings of integers of
number fields. \n\nIn recent years\, techniques from arithmetic geometry
have been used extensively to attack this problem. One such instance is th
e work of García-Fritz and Pasten (from 2019) which showed that the analo
gue of Hilbert's 10th problem is unsolvable in the ring of integers of num
ber fields of the form $\\mathbb{Q}(\\sqrt[3]{p}\,\\sqrt{-q})$ for positiv
e proportions of primes $p$ and $q$. In joint work with Lei and Sprung\, w
e improve their proportions and extend their results in several directions
. We achieve this by using multiple elliptic curves\, and by replacing the
ir Iwasawa theory arguments by a more direct method.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elchin Hasanalizade (University of Lethbridge)
DTSTART:20221017T180000Z
DTEND:20221017T190000Z
DTSTAMP:20260404T095423Z
UID:NTC/7
DESCRIPTION:Title: Sums of Fibonacci numbers close to a power of $2$\nby Elchin Hasana
lizade (University of Lethbridge) as part of Lethbridge number theory and
combinatorics seminar\n\nLecture held in University of Lethbridge: M1040 (
Markin Hall).\n\nAbstract\nThe Fibonacci sequence $(F_n)_{n \\geq 0}$ is t
he binary recurrence sequence defined by $F_0 = F_1 = 1$ and\n$$\nF_{n+2}
= F_{n+1} + F_n \\text{ for all } n \\geq 0.\n$$\nThere is a broad litera
ture on the Diophantine equations involving the Fibonacci numbers. In this
talk\, we will study the Diophantine inequality\n$$\n| F_n + F_m - 2^a |
< 2^{a/2}\n$$\nin positive integers $n\, m$ and $a$ with $n \\geq m$. The
main tools used are lower bounds for linear forms in logarithms due to Mat
veev and Dujella-Pethö version of the Baker-Davenport reduction method in
Diophantine approximation.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dave Morris (University of Lethbridge)
DTSTART:20221024T180000Z
DTEND:20221024T190000Z
DTSTAMP:20260404T095423Z
UID:NTC/8
DESCRIPTION:Title: On vertex-transitive graphs with a unique Hamiltonian circle\nby Da
ve Morris (University of Lethbridge) as part of Lethbridge number theory a
nd combinatorics seminar\n\nLecture held in University of Lethbridge: M104
0 (Markin Hall).\n\nAbstract\nWe will discuss graphs that have a unique Ha
miltonian cycle and are vertex-transitive\, which means there is an automo
rphism that takes any vertex to any other vertex. Cycles are the only exam
ples with finitely many vertices\, but the situation is more interesting f
or infinite graphs. (Infinite graphs do not have ``Hamiltonian cycles''\,
but there are natural analogues.) The case where the graph has only finite
ly many ends is not difficult\, but we do not know whether there are examp
les with infinitely many ends. This is joint work in progress with Bobby M
iraftab.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Solaleh Bolvardizadeh (University of Lethbridge)
DTSTART:20221121T190000Z
DTEND:20221121T200000Z
DTSTAMP:20260404T095423Z
UID:NTC/9
DESCRIPTION:Title: On the Quality of the $ABC$-Solutions\nby Solaleh Bolvardizadeh (Un
iversity of Lethbridge) as part of Lethbridge number theory and combinator
ics seminar\n\nLecture held in University of Lethbridge: M1040 (Markin Hal
l).\n\nAbstract\nThe quality of the triplet $(a\,b\,c)$\, where $\\gcd(a\,
b\,c) = 1$\, satisfying $a + b = c$ is defined as\n$$\nq(a\,b\,c) = \\frac
{\\max\\{\\log |a|\, \\log |b|\, \\log |c|\\}}{\\log \\mathrm{rad}(|abc|)}
\,\n$$\nwhere $\\mathrm{rad}(|abc|)$ is the product of distinct prime fact
ors of $|abc|$. We call such a triplet an $ABC$-solution. The $ABC$-conjec
ture states that given $\\epsilon > 0$ the number of the $ABC$-solutions $
(a\,b\,c)$ with $q(a\,b\,c) \\geq 1 + \\epsilon$ is finite.\n\nIn the firs
t part of this talk\, under the $ABC$-conjecture\, we explore the quality
of certain families of the $ABC$-solutions formed by terms in Lucas and as
sociated Lucas sequences. We also introduce\, unconditionally\, a new fami
ly of $ABC$-solutions that has quality $> 1$.\n\nIn the remaining of the t
alk\, we prove a conjecture of Erd\\"os on the solutions of the Brocard-Ra
manujan equation\n$$\nn! + 1 = m^2\n$$\nby assuming an explicit version of
the $ABC$-conjecture proposed by Baker.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Douglas Ulmer (University of Arizona)
DTSTART:20230327T180000Z
DTEND:20230327T190000Z
DTSTAMP:20260404T095423Z
UID:NTC/10
DESCRIPTION:Title: $p$-torsion of Jacobians for unramified $\\mathbb{Z}/p\\mathbb{Z}$-cov
ers of curves\nby Douglas Ulmer (University of Arizona) as part of Let
hbridge number theory and combinatorics seminar\n\nLecture held in Univers
ity of Lethbridge: M1040 (Markin Hall).\n\nAbstract\nIt is a classical pro
blem to understand the set of Jacobians of curves\namong all abelian varie
ties\, i.e.\, the image of the map $M_g\\to A_g$\nwhich sends a curve $X$
to its Jacobian $J_X$. In characteristic $p$\,\n$A_g$ has interesting fil
trations\, and we can ask how the image of\n$M_g$ interacts with them. Co
ncretely\, which groups schemes arise as\nthe p-torsion subgroup $J_X[p]$
of a Jacobian? We consider this\nproblem in the context of unramified $Z/
pZ$ covers $Y\\to X$ of curves\,\nasking how $J_Y[p]$ is related to $J_X[p
]$. Translating this into a\nproblem about de Rham cohmology yields some
results using\nclassical ideas of Chevalley and Weil. This is joint work
with Bryden\nCais.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Males (University of Manitoba)
DTSTART:20230320T180000Z
DTEND:20230320T190000Z
DTSTAMP:20260404T095423Z
UID:NTC/11
DESCRIPTION:Title: Forgotten conjectures of Andrews for Nahm-type sums\nby Joshua Mal
es (University of Manitoba) as part of Lethbridge number theory and combin
atorics seminar\n\nLecture held in University of Lethbridge: M1040 (Markin
Hall).\n\nAbstract\nIn his famous '86 paper\, Andrews made several conjec
tures on\nthe function $\\sigma(q)$ of Ramanujan\, including that it has\n
coefficients (which count certain partition-theoretic objects) whose\nsup
grows in absolute value\, and that it has infinitely many Fourier\ncoeffic
ients that vanish. These conjectures were famously proved by\nAndrews-Dyso
n-Hickerson in their '88 Invent. paper\, and the function\n$\\sigma$ has b
een related to the arithmetic of $\\mathbb{Z}[\\sqrt{6}]$\nby Cohen (and e
xtensions by Zwegers)\, and is an important first\nexample of quantum modu
lar forms introduced by Zagier.\n\nA closer inspection of Andrews' '86 pap
er reveals several more\nfunctions that have been a little left in the sha
dow of their sibling\n$\\sigma$\, but which also exhibit extraordinary beh
aviour. In an\nongoing project with Folsom\, Rolen\, and Storzer\, we stud
y the function\n$v_1(q)$ which is given by a Nahm-type sum and whose coeff
icients\ncount certain differences of partition-theoretic objects. We give
\nexplanations of four conjectures made by Andrews on $v_1$\, which\nrequi
re a blend of novel and well-known techniques\, and reveal that\n$v_1$ sho
uld be intimately linked to the arithmetic of the imaginary\nquadratic fie
ld $\\mathbb{Q}[\\sqrt{-3}]$.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristhian Garay (Centro de Investigación en Matemáticas (CIMAT)\
, Guanajuato)
DTSTART:20230206T190000Z
DTEND:20230206T200000Z
DTSTAMP:20260404T095423Z
UID:NTC/12
DESCRIPTION:Title: Generalized valuations and idempotization of schemes\nby Cristhian
Garay (Centro de Investigación en Matemáticas (CIMAT)\, Guanajuato) as
part of Lethbridge number theory and combinatorics seminar\n\nLecture held
in University of Lethbridge: M1040 (Markin Hall).\n\nAbstract\nClassical
valuation theory has proved to be a valuable tool in number theory\, algeb
raic geometry and singularity theory. For example\, one can enrich spectra
of rings with new points coming from valuations defined on them and takin
g values in totally ordered abelian groups.\n\n\n\nTotally ordered groups
are examples of idempotent semirings\, and generalized valuations appear w
hen we replace totally ordered abelian groups with more general idempotent
semirings. An important example of idempotent semiring is the tropical se
mifield. \n\n\nAs an application of this set of ideas\, we show how to ass
ociate an idempotent version of the structure sheaf of a scheme\, which be
haves particularly well with respect to idempotization of closed subscheme
s.\n\n\nThis is a joint work with Félix Baril Boudreau.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Renate Scheidler (University of Calgary)
DTSTART:20230313T180000Z
DTEND:20230313T190000Z
DTSTAMP:20260404T095423Z
UID:NTC/13
DESCRIPTION:Title: Orienteering on Supersingular Isogeny Volcanoes Using One Endomorphism
\nby Renate Scheidler (University of Calgary) as part of Lethbridge nu
mber theory and combinatorics seminar\n\nLecture held in University of Let
hbridge: M1040 (Markin Hall).\n\nAbstract\nElliptic curve isogeny path fin
ding has many applications in number theory and cryptography. For supersin
gular curves\, this problem is known to be easy when one small endomorphis
m or the entire endomorphism ring are known. Unfortunately\, computing the
endomorphism ring\, or even just finding one small endomorphism\, is hard
. How difficult is path finding in the presence of one (not necessarily s
mall) endomorphism? We use the volcano structure of the oriented supersing
ular isogeny graph to answer this question. We give a classical algorithm
for path finding that is subexponential in the degree of the endomorphism
and linear in a certain class number\, and a quantum algorithm for finding
a smooth isogeny (and hence also a path) that is subexponential in the di
scriminant of the endomorphism. A crucial tool for navigating supersingula
r oriented isogeny volcanoes is a certain class group action on oriented e
lliptic curves which generalizes the well-known class group action in the
setting of ordinary elliptic curves.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Youness Lamzouri (Institut Élie Cartan de Lorraine (IECL) of the
Université de Lorraine in Nancy)
DTSTART:20230109T190000Z
DTEND:20230109T200000Z
DTSTAMP:20260404T095423Z
UID:NTC/14
DESCRIPTION:Title: A walk on Legendre paths\nby Youness Lamzouri (Institut Élie Cart
an de Lorraine (IECL) of the Université de Lorraine in Nancy) as part of
Lethbridge number theory and combinatorics seminar\n\nLecture held in Univ
ersity of Lethbridge: M1040 (Markin Hall).\n\nAbstract\nThe Legendre symbo
l is one of the most basic\, mysterious and extensively studied objects in
number theory. It is a multiplicative function that encodes information a
bout whether an integer is a square modulo an odd prime $p$. The Legendre
symbol was introduced by Adrien-Marie Legendre in 1798\, and has since fou
nd countless applications in various areas of mathematics as well as in ot
her fields including cryptography. In this talk\, we shall explore what we
call ``Legendre paths''\, which encode information about the values of th
e Legendre symbol. The Legendre path modulo $p$ is defined as the polygona
l path in the plane formed by joining the partial sums of the Legendre sym
bol modulo $p$. In particular\, we will attempt to answer the following qu
estions as we vary over the primes $p$: how are these paths distributed? h
ow do their maximums behave? and what proportion of the path is above the
real axis? Among our results\, we prove that these paths converge in law\,
in the space of continuous functions\, to a certain random Fourier series
constructed using Rademakher random multiplicative functions. Part of thi
s work is joint with Ayesha Hussain.\n\nThis talk is part of the PIMS Dist
inguished Speaker Series. The registration link is only valid for this tal
k.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antonella Perucca (University of Luxembourg)
DTSTART:20230123T163000Z
DTEND:20230123T173000Z
DTSTAMP:20260404T095423Z
UID:NTC/15
DESCRIPTION:Title: Recent advances in Kummer theory\nby Antonella Perucca (University
of Luxembourg) as part of Lethbridge number theory and combinatorics semi
nar\n\n\nAbstract\nKummer theory is a classical theory about radical exten
sions of fields in the case where suitable roots of unity are present in t
he base field. Motivated by problems close to Artin's primitive root conje
cture\, we have investigated the degree of families of general Kummer exte
nsions of number fields\, providing parametric closed formulas. We present
a series of papers that are in part joint work with Christophe Debry\, Fr
itz Hörmann\, Pietro Sgobba\, and Sebastiano Tronto.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neelam Kandhil (The Institute of Mathematical Sciences (IMSc)\, Ch
ennai)
DTSTART:20230116T163000Z
DTEND:20230116T173000Z
DTSTAMP:20260404T095423Z
UID:NTC/16
DESCRIPTION:Title: On linear independence of Dirichlet L-values\nby Neelam Kandhil (T
he Institute of Mathematical Sciences (IMSc)\, Chennai) as part of Lethbri
dge number theory and combinatorics seminar\n\n\nAbstract\nIt is an open q
uestion of Baker whether the Dirichlet L-values at 1 with fixed modulus ar
e linearly\nindependent over the rational numbers. The best-known result i
s due to Baker\, Birch and Wirsing\, which affirms\nthis when the modulus
of the associated Dirichlet character is co-prime to its Euler's phi value
. In this talk\,\nwe will discuss an extension of this result to any arbit
rary family of moduli. The interplay between the\nresulting ambient number
fields brings new technical issues and complications hitherto absent in t
he context of\na fixed modulus. We will also investigate the linear indepe
ndence of such values at integers greater than 1.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oussama Hamza (University of Western Ontario)
DTSTART:20230130T190000Z
DTEND:20230130T200000Z
DTSTAMP:20260404T095423Z
UID:NTC/17
DESCRIPTION:Title: Filtrations\, arithmetic and explicit examples in an equivariant conte
xt\nby Oussama Hamza (University of Western Ontario) as part of Lethbr
idge number theory and combinatorics seminar\n\nLecture held in M1040 (Mar
kin Hall).\n\nAbstract\nPro-$p$ groups arise naturally in number theory as
quotients of absolute Galois groups over number fields. These groups are
quite mysterious. During the 60's\, Koch gave a presentation of some of th
ese quotients. Furthermore\, around the same period\, Jennings\, Golod\, S
hafarevich and Lazard introduced two integer sequences $(a_n)$ and $(c_n)$
\, closely related to a special filtration of a finitely generated pro-p g
roup $G$\, called the Zassenhaus filtration. These sequences give the card
inality of $G$\, and characterize its topology. For instance\, we have the
well-known Gocha's alternative (Golod and Shafarevich): There exists an i
nteger $n$ such that $a_n=0$ (or $c_n$ has a polynomial growth) if and onl
y if $G$ is a Lie group over $p$-adic fields.\n\nIn 2016\, Minac\, Rogelst
ad and Tan inferred an explicit relation between $a_n$ and $c_n$. Recently
(2022)\, considering geometrical ideas of Filip and Stix\, Hamza got more
precise relations in an equivariant context: when the automorphism group
of $G$ admits a subgroup of order a prime $q$ dividing $p-1$.\n\nIn this t
alk\, we present equivariant relations inferred by Hamza (2022) and give e
xplicit examples in an arithmetical context.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florent Jouve (Université de Bordeaux)
DTSTART:20230227T163000Z
DTEND:20230227T173000Z
DTSTAMP:20260404T095423Z
UID:NTC/18
DESCRIPTION:Title: Fluctuations in the distribution of Frobenius automorphisms in number
field extensions\nby Florent Jouve (Université de Bordeaux) as part o
f Lethbridge number theory and combinatorics seminar\n\n\nAbstract\nGiven
a Galois extension of number fields $L/K$\, the Chebotarev Density Theorem
asserts that\, away from ramified primes\, Frobenius automorphisms equidi
stribute in the set of conjugacy classes of ${\\rm Gal}(L/K)$. In this tal
k we report on joint work with D. Fiorilli in which we study the variation
s of the error term in Chebotarev’s Theorem as $L/K$ runs over certain f
amilies of extensions. We shall explain some consequences of this analysis
: regarding first "Linnik type problems" on the least prime ideal in a giv
en Frobenius set\, and second\, the existence of unconditional "Chebyshev
biases" in the context of number fields. Time permitting we will mention j
oint work with R. de La Bretèche and D. Fiorilli in which we go one step
further and study moments of the distribution of Frobenius automorphisms.\
n
LOCATION:https://stable.researchseminars.org/talk/NTC/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Voight (Dartmouth College)
DTSTART:20230306T190000Z
DTEND:20230306T200000Z
DTSTAMP:20260404T095423Z
UID:NTC/19
DESCRIPTION:Title: A norm refinement of Bezout's Lemma\, and quaternion orders\nby Jo
hn Voight (Dartmouth College) as part of Lethbridge number theory and comb
inatorics seminar\n\nLecture held in University of Lethbridge: M1040 (Mark
in Hall).\n\nAbstract\nGiven coprime integers a\,b\, the classical identit
y of Bezout provides\nintegers u\,v such that au-bv = 1. We consider refi
nements to this\nidentity\, where we ask that u\,v are norms from a quadra
tic extension.\nWe then find ourselves counting optimal embeddings of a qu
adratic\norder in a quaternion order\, for which we give explicit formulas
in\nmany cases. This is joint work with Donald Cartwright and Xavier\nRo
ulleau.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristhian Garay (Centro de Investigación en Matemáticas (CIMAT)\
, Guanajuato)
DTSTART:20230206T221000Z
DTEND:20230206T234500Z
DTSTAMP:20260404T095423Z
UID:NTC/20
DESCRIPTION:Title: An invitation to the algebraic geometry over idempotent semirings (Lec
ture 1 of 2)\nby Cristhian Garay (Centro de Investigación en Matemát
icas (CIMAT)\, Guanajuato) as part of Lethbridge number theory and combina
torics seminar\n\nLecture held in University of Lethbridge: B716 (Universi
ty Hall).\n\nAbstract\nIdempotent semirings have been relevant in several
branches of applied mathematics\, like formal languages and combinatorial
optimization.\n\n\nThey were brought recently to pure mathematics thanks t
o its link with tropical geometry\, which is a relatively new branch of ma
thematics that has been useful in solving some problems and conjectures in
classical algebraic geometry. \n\n\nHowever\, up to now we do not have a
proper algebraic formalization of what could be called “Tropical Algebra
ic Geometry”\, which is expected to be the geometry arising from idempot
ent semirings. \n\n\nIn this mini course we aim to motivate the necessity
for such theory\, and we recast some old constructions in order theory in
terms of commutative algebra of semirings and modules over them.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristhian Garay (Centro de Investigación en Matemáticas (CIMAT)\
, Guanajuato)
DTSTART:20230209T221000Z
DTEND:20230209T234500Z
DTSTAMP:20260404T095423Z
UID:NTC/21
DESCRIPTION:Title: An invitation to the algebraic geometry over idempotent semirings (Lec
ture 2 of 2)\nby Cristhian Garay (Centro de Investigación en Matemát
icas (CIMAT)\, Guanajuato) as part of Lethbridge number theory and combina
torics seminar\n\nLecture held in University of Lethbridge: B716 (Universi
ty Hall).\n\nAbstract\nIdempotent semirings have been relevant in several
branches of applied mathematics\, like formal languages and combinatorial
optimization.\n\n\nThey were brought recently to pure mathematics thanks t
o its link with tropical geometry\, which is a relatively new branch of ma
thematics that has been useful in solving some problems and conjectures in
classical algebraic geometry. \n\n\nHowever\, up to now we do not have a
proper algebraic formalization of what could be called “Tropical Algebra
ic Geometry”\, which is expected to be the geometry arising from idempot
ent semirings. \n\n\nIn this mini course we aim to motivate the necessity
for such theory\, and we recast some old constructions in order theory in
terms of commutative algebra of semirings and modules over them.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harald Andrés Helfgott (University of Göttingen/Institut de Math
ématiques de Jussieu)
DTSTART:20230403T163000Z
DTEND:20230403T173000Z
DTSTAMP:20260404T095423Z
UID:NTC/22
DESCRIPTION:Title: Expansion\, divisibility and parity\nby Harald Andrés Helfgott (U
niversity of Göttingen/Institut de Mathématiques de Jussieu) as part of
Lethbridge number theory and combinatorics seminar\n\nLecture held in Univ
ersity of Lethbridge: M1040 (Markin Hall).\n\nAbstract\nWe will discuss a
graph that encodes the divisibility properties of integers by primes. We p
rove that this graph has a strong local expander property almost everywher
e. We then obtain several consequences in number theory\, beyond the tradi
tional parity barrier\, by combining our result with Matomaki-Radziwill. F
or instance: for lambda the Liouville function (that is\, the completely m
ultiplicative function with $\\lambda(p) = -1$ for every prime)\, $(1/\\lo
g x) \\sum_{n\\leq x} \\lambda(n) \\lambda(n+1)/n = O(1/\\sqrt(\\log \\log
x))$\, which is stronger than well-known results by Tao and Tao-Teravaine
n. We also manage to prove\, for example\, that $\\lambda(n+1)$ averages t
o $0$ at almost all scales when $n$ restricted to have a specific number o
f prime divisors $\\Omega(n)=k$\, for any "popular" value of $k$ (that is\
, $k = \\log \\log N + O(\\sqrt(\\log \\log N)$) for $n \\leq N$).\n
LOCATION:https://stable.researchseminars.org/talk/NTC/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kelly Emmrich (Colorado State University)
DTSTART:20230213T190000Z
DTEND:20230213T200000Z
DTSTAMP:20260404T095423Z
UID:NTC/23
DESCRIPTION:Title: The principal Chebotarev density theorem\nby Kelly Emmrich (Colora
do State University) as part of Lethbridge number theory and combinatorics
seminar\n\nLecture held in University of Lethbridge: M1040 (Markin Hall).
\n\nAbstract\nLet K/k be a finite Galois extension. We define a principal
version of the Chebotarev density theorem which represents the density of
prime ideals of k that factor into a product of principal prime ideals in
K. We find explicit equations to express the principal density in terms of
the invariants of K/k and give an effective bound which can be used to ve
rify the non-splitting of the Hilbert exact sequence.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No talk - Reading Week
DTSTART:20230220T190000Z
DTEND:20230220T200000Z
DTSTAMP:20260404T095423Z
UID:NTC/24
DESCRIPTION:by No talk - Reading Week as part of Lethbridge number theory
and combinatorics seminar\n\nLecture held in University of Lethbridge: M10
40 (Markin Hall).\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/NTC/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriel Verret (University of Auckland\, New Zealand)
DTSTART:20230919T200000Z
DTEND:20230919T210000Z
DTSTAMP:20260404T095423Z
UID:NTC/25
DESCRIPTION:Title: Vertex-transitive graphs with large automorphism groups\nby Gabrie
l Verret (University of Auckland\, New Zealand) as part of Lethbridge numb
er theory and combinatorics seminar\n\nLecture held in University of Lethb
ridge: M1060 (Markin Hall).\n\nAbstract\nMany results in algebraic graph t
heory can be viewed as upper bounds on the size of the automorphism group
of graphs satisfying various hypotheses. These kinds of results have many
applications. For example\, Tutte's classical theorem on 3-valent arc-tran
sitive graphs led to many other important results about these graphs\, inc
luding enumeration\, both of small order and in the asymptotical sense. Th
is naturally leads to trying to understand barriers to this type of result
s\, namely graphs with large automorphism groups. We will discuss this\, e
specially in the context of vertex-transitive graphs of fixed valency. We
will highlight the apparent dichotomy between graphs with automorphism gro
up of polynomial (with respect to the order of the graph) size\, versus on
es with exponential size.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sedanur Albayrak (University of Calgary)
DTSTART:20230926T200000Z
DTEND:20230926T210000Z
DTSTAMP:20260404T095423Z
UID:NTC/26
DESCRIPTION:Title: Quantitative Estimates for the Size of an Intersection of Sparse Autom
atic Sets\nby Sedanur Albayrak (University of Calgary) as part of Leth
bridge number theory and combinatorics seminar\n\nLecture held in Universi
ty of Lethbridge: M1060 (Markin Hall).\n\nAbstract\nIn 1979\, Erdős conje
ctured that for $k \\geq 9$\, $2^k$ is not the sum of distinct powers of $
3$. That is\, the set of powers of two (which is $2$-automatic) and the $3
$-automatic set consisting of numbers\nwhose ternary expansions omit $2$ h
as finite intersection. In the theory of automata\, a theorem of Cobham (1
969) says that if $k$ and $\\ell$ are two multiplicatively independent nat
ural numbers then a subset of the natural numbers that is both $k$- and $\
\ell$-automatic is eventually periodic. A multidimensional extension was l
ater given by Semenov (1977). Motivated by Erdős' conjecture and in light
of Cobham’s theorem\, we give a quantitative version of the Cobham-Seme
nov theorem for sparse automatic sets\, showing that the intersection of a
sparse $k$-automatic subset of $\\mathbb{N}^d$ and a sparse $\\ell$-autom
atic subset of $\\mathbb{N}^d$ is finite. Moreover\, we give effectively c
omputable upper bounds on the size of the intersection in terms of data fr
om the automata that accept these sets.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kübra Benli (University of Lethbridge)
DTSTART:20231003T200000Z
DTEND:20231003T210000Z
DTSTAMP:20260404T095423Z
UID:NTC/27
DESCRIPTION:Title: Sums of proper divisors with missing digits\nby Kübra Benli (Univ
ersity of Lethbridge) as part of Lethbridge number theory and combinatoric
s seminar\n\nLecture held in University of Lethbridge: M1060 (Markin Hall)
.\n\nAbstract\nIn 1992\, Erdős\, Granville\, Pomerance\, and Spiro conjec
tured that if $\\mathcal{A}$ is a set of integers with asymptotic density
zero then the preimage of $\\mathcal{A}$ under $s(n)$\, sum-of-proper-divi
sors function\, also has asymptotic density zero. In this talk\, we will d
iscuss the verification of this conjecture when $\\mathcal{A}$ is taken to
be the set of integers with missing digits (also known as ellipsephic int
egers) by giving a quantitative estimate on the size of the set $s^{-1}(\\
mathcal{A})$. This is joint work with Giulia Cesana\, Cécile Dartyge\, Ch
arlotte Dombrowsky and Lola Thompson.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wanlin Li (Washington University in St. Louis)
DTSTART:20231012T200000Z
DTEND:20231012T210000Z
DTSTAMP:20260404T095423Z
UID:NTC/28
DESCRIPTION:Title: Basic reductions of abelian varieties\nby Wanlin Li (Washington Un
iversity in St. Louis) as part of Lethbridge number theory and combinatori
cs seminar\n\nLecture held in University of Lethbridge: M1060 (Markin Hall
).\n\nAbstract\nGiven an abelian variety $A$ defined over a number field\,
a conjecture attributed to Serre states\nthat the set of primes at which
$A$ admits ordinary reduction is of positive density. This conjecture had
been proved for elliptic curves (Serre\, 1977)\, abelian surfaces (Katz 19
82\, Sawin 2016) and certain higher dimensional abelian varieties (Pink 19
83\, Fite 2021\, etc).\n\nIn this talk\, we will discuss ideas behind thes
e results and recent progress for abelian varieties with non-trivial endom
orphisms\, including the case where $A$ has almost complex multiplication
by an abelian CM field\, based on joint work with Cantoral-Farfan\, Mantov
an\, Pries\, and Tang.\n\nApart from ordinary reduction\, we will also dis
cuss the set of primes at which an abelian variety admits basic reduction\
, generalizing a result of Elkies on the infinitude of supersingular prime
s for elliptic curves. This is joint work with Mantovan\, Pries\, and Tang
.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhenchao Ge (University of Waterloo)
DTSTART:20231017T200000Z
DTEND:20231017T210000Z
DTSTAMP:20260404T095423Z
UID:NTC/29
DESCRIPTION:Title: A Weyl-type inequality for irreducible elements in function fields\, w
ith applications\nby Zhenchao Ge (University of Waterloo) as part of L
ethbridge number theory and combinatorics seminar\n\nLecture held in Unive
rsity of Lethbridge: M1060 (Markin Hall).\n\nAbstract\nWe establish a Weyl
-type estimate for exponential sums over irreducible elements in function
fields. As an application\, we generalize an equidistribution theorem of R
hin. Our estimate works for polynomials with degree higher than the charac
teristic of the field\, a barrier to the traditional Weyl differencing met
hod. In this talk\, we briefly introduce Lê-Liu-Wooley’s original argum
ent for ordinary Weyl sums (taken over all elements)\, and how we generali
ze it to estimate bilinear exponential sums with general coefficients. Thi
s is joint work with Jérémy Campagne (Waterloo)\, Thái Hoàng Lê\n(Mis
sissippi) and Yu-Ru Liu (Waterloo).\n
LOCATION:https://stable.researchseminars.org/talk/NTC/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yu-Ru Liu (University of Waterloo)
DTSTART:20231025T200000Z
DTEND:20231025T210000Z
DTSTAMP:20260404T095423Z
UID:NTC/30
DESCRIPTION:Title: Fermat vs Waring: an introduction to number theory in function fields<
/a>\nby Yu-Ru Liu (University of Waterloo) as part of Lethbridge number th
eory and combinatorics seminar\n\nLecture held in University of Lethbridge
: M1040 (Markin Hall).\n\nAbstract\nLet $\\Z$ be the ring of integers\, an
d let $\\mathbb{F}_p[t]$ be the ring of polynomials in one variable define
d over the finite field $\\mathbb{F}_p$ of $p$ elements. Since the charact
eristic of $\\Z$ is $0$\, while that of $\\mathbb{F}_p[t]$ is the positive
prime number $p$\, it is a striking theme in arithmetic that these two ri
ngs faithfully resemble each other. The study of the similarity and differ
ence between $\\Z$ and $\\mathbb{F}_p[t]$ lies in the field that relates n
umber fields to function fields. In this talk\, we will investigate some D
iophantine problems in the settings of $\\Z$ and $\\mathbb{F}_p[t]$\, incl
uding Fermat's Last Theorem and Waring's problem.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joy Morris (University of Lethbridge)
DTSTART:20231031T200000Z
DTEND:20231031T210000Z
DTSTAMP:20260404T095423Z
UID:NTC/31
DESCRIPTION:Title: Easy Detection of (Di)Graphical Regular Representations\nby Joy Mo
rris (University of Lethbridge) as part of Lethbridge number theory and co
mbinatorics seminar\n\nLecture held in University of Lethbridge: M1060 (Ma
rkin Hall).\n\nAbstract\nGraphical and Digraphical Regular Representations
(GRRs and DRRs) are a concrete way to visualise the regular action of a g
roup\, using graphs. More precisely\, a GRR or DRR on the group $G$ is a (
di)graph whose automorphism group is isomorphic to the regular action of $
G$ on itself by right-multiplication.\n\nFor a (di)graph to be a DRR or GR
R on $G$\, it must be a Cayley (di)graph on $G$. Whenever the group $G$ ad
mits an automorphism that fixes the connection set of the Cayley (di)graph
setwise\, this induces a nontrivial graph automorphism that fixes the ide
ntity vertex\, which means that the (di)graph is not a DRR or GRR. Checkin
g whether or not there is any group automorphism that fixes a particular c
onnection set can be done very quickly and easily compared with checking w
hether or not any nontrivial graph automorphism fixes some vertex\, so it
would be nice to know if there are circumstances under which the simpler t
est is enough to guarantee whether or not the Cayley graph is a GRR or DRR
. I will present a number of results on this question.\n\nThis is based on
joint work with Dave Morris and with Gabriel Verret.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abbas Maarefparvar (University of Lethbridge)
DTSTART:20231107T210000Z
DTEND:20231107T220000Z
DTSTAMP:20260404T095423Z
UID:NTC/32
DESCRIPTION:Title: Some Pólya fields of small degrees\nby Abbas Maarefparvar (Univer
sity of Lethbridge) as part of Lethbridge number theory and combinatorics
seminar\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).\
n\nAbstract\nHistorically\, the notion of Pólya fields dates back to some
works of George Pólya and Alexander Ostrowski\, in 1919\, on entire func
tions with integervalues at integers\; a number field $K$ with ring of in
tegers $\\mathcal{O}_K$ is called a Pólya field whenever the $\\mathcal{
O}_K$-module $\\{ f \\in K[X] : f(\\mathcal{O}_K ) \\subseteq \\mathcal{O
}_K \\}$ admits an $\\mathcal{O}_K$-basis with exactly one member from eac
h degree. Pólya fields can be thought of as a generalization of number fi
elds with class number one\, and their classification of a specific degree
has become recently an active research subject in algebraic number theory
. In this talk\, I will present some criteria for $K$ to be a Pólya field
. Then I will give some results concerning Pólya fields of degrees $2\, 3
$\, and $6$.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sreerupa Bhattacharjee (University of Lethbridge)
DTSTART:20231121T210000Z
DTEND:20231121T220000Z
DTSTAMP:20260404T095423Z
UID:NTC/34
DESCRIPTION:Title: A Survey of Büthe's Method for Estimating Prime Counting Functions\nby Sreerupa Bhattacharjee (University of Lethbridge) as part of Lethbri
dge number theory and combinatorics seminar\n\nLecture held in University
of Lethbridge: M1060 (Markin Hall).\n\nAbstract\nThis talk will begin with
a study on explicit bounds for $\\psi(x)$ starting with the work of Rosse
r in 1941. It will also cover various improvements over the years includin
g the works of Rosser and Schoenfeld\, Dusart\, Faber-Kadiri\, Platt-Trudg
ian\, Büthe\, and Fiori-Kadiri-Swidinsky. In the second part of this talk
\, I will provide an overview of my master's thesis which is a survey on `
Estimating $\\pi(x)$ and Related Functions under Partial RH Assumptions' b
y Jan Büthe. This article provides the best known bounds for $\\psi(x)$ f
or small values of~$x$ in the interval $[e^{50}\,e^{3000}]$. A distinctive
feature of this paper is the use of Logan's function and its Fourier Tran
sform. I will be presenting the main theorem in Büthe's paper regarding e
stimates for $\\psi(x)$ with other necessary results required to understan
d the proof.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ha Tran (Concordia University of Edmonton)
DTSTART:20231128T210000Z
DTEND:20231128T220000Z
DTSTAMP:20260404T095423Z
UID:NTC/35
DESCRIPTION:Title: The Size Function For Imaginary Cyclic Sextic Fields\nby Ha Tran (
Concordia University of Edmonton) as part of Lethbridge number theory and
combinatorics seminar\n\nLecture held in University of Lethbridge: M1060 (
Markin Hall).\n\nAbstract\nThe size function $h^0$ for a number field is a
nalogous to the dimension of the\nRiemann-Roch spaces of divisors on an al
gebraic curve. Van der Geer and Schoof conjectured\nthat $h^0$ attains its
maximum at the trivial class of Arakelov divisors if that field is Galois
over\n$\\mathbb{Q}$ or over an imaginary quadratic field. This conjecture
was proved for all number fields with the unit group of rank $0$ and $1$\
, and also for cyclic cubic fields which have unit group of rank\ntwo. In
this talk\, we will discuss the main idea to prove that the conjecture als
o holds for\ntotally imaginary cyclic sextic fields\, another class of num
ber fields with unit group of rank\ntwo. This is joint work with Peng Tian
and Amy Feaver.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hadi Kharaghani (University of Lethbridge)
DTSTART:20240124T204500Z
DTEND:20240124T214500Z
DTSTAMP:20260404T095423Z
UID:NTC/36
DESCRIPTION:Title: Projective Planes and Hadamard Matrices\nby Hadi Kharaghani (Unive
rsity of Lethbridge) as part of Lethbridge number theory and combinatorics
seminar\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).
\n\nAbstract\nIt is conjectured that there is no projective plane of order
12. Balanced splittable\nHadamard matrices were introduced in 2018. In 20
23\, it was shown that a projective\nplane of order 12 is equivalent to a
balanced multi-splittable Hadamard matrix of\norder 144. There will be an
attempt to show the equivalence in a way that may\nrequire little backgrou
nd.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Félix Baril Boudreau (University of Lethbridge)
DTSTART:20240229T204500Z
DTEND:20240229T214500Z
DTSTAMP:20260404T095423Z
UID:NTC/37
DESCRIPTION:Title: The Distribution of Logarithmic Derivatives of Quadratic L-functions i
n Positive Characteristic\nby Félix Baril Boudreau (University of Let
hbridge) as part of Lethbridge number theory and combinatorics seminar\n\n
Lecture held in University of Lethbridge: M1040 (Markin Hall).\n\nAbstract
\nTo each square-free monic polynomial $D$ in a fixed polynomial ring $\\m
athbb{F}_q[t]$\, we can associate a real quadratic character $\\chi_D$\, a
nd then a Dirichlet $L$-function $L(s\,\\chi_D)$. We compute the limiting
distribution of the family of values $L'(1\,\\chi_D)/L(1\,\\chi_D)$ as $D$
runs through the square-free monic polynomials of $\\mathbb{F}_q[t]$ and
establish that this distribution has a smooth density function. Time permi
tting\, we discuss connections of this result with Euler-Kronecker constan
ts and ideal class groups of quadratic extensions. This is joint work with
Amir Akbary.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sho Suda (National Defense Academy of Japan)
DTSTART:20240313T194500Z
DTEND:20240313T204500Z
DTSTAMP:20260404T095423Z
UID:NTC/38
DESCRIPTION:Title: On extremal orthogonal arrays\nby Sho Suda (National Defense Acade
my of Japan) as part of Lethbridge number theory and combinatorics seminar
\n\nLecture held in University of Lethbridge: M1060 (Markin Hall).\n\nAbst
ract\nAn orthogonal array with parameters $(N\,n\,q\,t)$ ($OA(N\,n\,q\,t)$
for short) is an $N\\times n$ matrix with entries from the alphabet $\\{1
\,2\,...\,q\\}$ such that in any its $t$ columns\, all possible row vector
s of length $t$ occur equally often. \nRao showed the following lower boun
d on $N$ for $OA(N\,n\,q\,2e)$: \n\\[\nN\\geq \\sum_{k=0}^e \\binom{n}{k}(
q-1)^k\, \n\\]\nand an orthogonal array is said to be complete or tight if
it achieves equality in this bound. \nIt is known by Delsarte (1973) that
for complete orthogonal arrays $OA(N\,n\,q\,2e)$\, the number of Hamming
distances between distinct two rows is $e$. \nOne of the classical problem
s is to classify complete orthogonal arrays. \n\nWe call an orthogonal ar
ray $OA(N\,n\,q\,2e-1)$ extremal if the number of Hamming distances betwee
n distinct two rows is $e$. \nIn this talk\, we review the classification
problem of complete orthogonal arrays with our contribution to the case $
t=4$ and show how to extend it to extremal orthogonal arrays. \nMoreover\,
we give a result for extremal orthogonal arrays which is a counterpart of
a result in block designs by Ionin and Shrikhande in 1993.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ertan Elma (University of Lethbridge)
DTSTART:20240131T204500Z
DTEND:20240131T214500Z
DTSTAMP:20260404T095423Z
UID:NTC/39
DESCRIPTION:Title: A Discrete Mean Value of the Riemann Zeta Function and its Derivatives
\nby Ertan Elma (University of Lethbridge) as part of Lethbridge numbe
r theory and combinatorics seminar\n\nLecture held in University of Lethbr
idge: M1060 (Markin Hall).\n\nAbstract\nIn this talk\, we will discuss an
estimate for a discrete mean value of the Riemann zeta function and its de
rivatives multiplied by Dirichlet polynomials. Assuming the Riemann Hypoth
esis\, we obtain a lower bound for the 2kth moment of all the derivatives
of the Riemann zeta function evaluated at its nontrivial zeros. This is ba
sed on a joint work with Kübra Benli and Nathan Ng.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samprit Ghosh (University of Calgary)
DTSTART:20240207T204500Z
DTEND:20240207T214500Z
DTSTAMP:20260404T095423Z
UID:NTC/40
DESCRIPTION:Title: Moments of higher derivatives related to Dirichlet L-functions\nby
Samprit Ghosh (University of Calgary) as part of Lethbridge number theory
and combinatorics seminar\n\nLecture held in University of Lethbridge: M1
060 (Markin Hall).\n\nAbstract\nThe distribution of values of Dirichlet $L
$-functions $L(s\, \\chi)$ for variable $\\chi$ has been studied extensiv
ely and has a vast literature. Moments of higher derivatives has been stud
ied as well\, by Soundarajan\, Sono\, Heath-Brown etc. However\, the stud
y of the same for the logarithmic derivative $L’(s\, \\chi)/ L(s\, \\chi
)$ is much more recent and was initiated by Ihara\, Murty etc. In this tal
k we will discuss higher derivatives of the logarithmic derivative and pre
sent some new results related to their distribution and moments at $s=1$.\
n
LOCATION:https://stable.researchseminars.org/talk/NTC/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abbas Maarefparvar (University of Lethbridge)
DTSTART:20240214T204500Z
DTEND:20240214T214500Z
DTSTAMP:20260404T095423Z
UID:NTC/41
DESCRIPTION:Title: Hilbert Class Fields and Embedding Problems\nby Abbas Maarefparvar
(University of Lethbridge) as part of Lethbridge number theory and combin
atorics seminar\n\nLecture held in University of Lethbridge: M1060 (Markin
Hall).\n\nAbstract\nThe class number one problem is one of the central su
bjects in algebraic number theory that turns back to the time of Gauss. Th
is problem has led to the classical embedding problem which asks whether o
r not any number field K can be embedded in a finite extension L with clas
s number one. Although Golod and Shafarevich gave a counterexample for the
classical embedding problem\, yet one may ask about the embedding in 'Pol
ya fields'\, a special generalization of class number one number fields. T
he latter is the 'new embedding problem' investigated by Leriche in 2014.\
nIn this talk\, I briefly review some well-known results in the literature
on the embedding problems. Then\, I will present the 'relativized' versio
n of the new embedding problem studied in a joint work with Ali Rajaei.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Fiori (University of Lethbridge)
DTSTART:20240306T204500Z
DTEND:20240306T214500Z
DTSTAMP:20260404T095423Z
UID:NTC/42
DESCRIPTION:Title: Tight approximation of sums over zeros of L-functions\nby Andrew F
iori (University of Lethbridge) as part of Lethbridge number theory and co
mbinatorics seminar\n\nLecture held in University of Lethbridge: M1060 (Ma
rkin Hall).\n\nAbstract\nIn various contexts explicit formula’s relate s
ums over primes (eg: numbers or ideals) to sums over zeros of some corresp
onding L-function(s). The aim of this talk is to explain how we tightly ap
proximate these sums over zeros in the context where one has zero free reg
ions and zero density results for the corresponding L-function(s) and how
we use this to get essentially best possible bounds for the error term in
the prime number theorem.\n\nThis talk discusses joint work with Habiba Ka
diri and Joshua Swidinsky as well as ongoing work with Mikko Jaskari and N
izar Bou Ezz.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Dijols (University of British Columbia)
DTSTART:20240320T194500Z
DTEND:20240320T204500Z
DTSTAMP:20260404T095423Z
UID:NTC/44
DESCRIPTION:Title: Parabolically induced representations of p-adic $G_2$ distinguished by
$SO_4$\nby Sarah Dijols (University of British Columbia) as part of L
ethbridge number theory and combinatorics seminar\n\nLecture held in Unive
rsity of Lethbridge: M1060 (Markin Hall).\n\nAbstract\nI will explain how
the Geometric Lemma allows us to classify parabolically induced representa
tions of the p-adic group $G_2$ distinguished by $SO_4$. In particular\, I
will describe a new approach\, in progress\, where we use the structure o
f the p-adic octonions and their quaternionic subalgebras to describe the
double coset space $P \\setminus G_2 / SO_4$\, where $P$ stands for the ma
ximal parabolic subgroups of $G_2$.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/44/
END:VEVENT
END:VCALENDAR